Lecture 4 (Fuzzy Set Operations)

Spring 2002Spring 2002 Lecture Lecture 0404 1صفحه

Lecture 4Lecture 4 (Fuzzy Set Operations)(Fuzzy Set Operations)

http://expertsys.4t.com

“We need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions”

L.A.Zadeh, 1962


Some points of the previous Some points of the previous lecturelecture

Fuzzy Logic is some kind of multi-Fuzzy Logic is some kind of multi-valued logic. Unlike crisp two valued valued logic. Unlike crisp two valued logic, the truth value in fuzzy logic logic, the truth value in fuzzy logic can be any number between 0 and 1 can be any number between 0 and 1 and hence is an extension to classical and hence is an extension to classical logiclogic

Consequently, many of the Natural Consequently, many of the Natural Language propositions can be Language propositions can be represented with fuzzy logicrepresented with fuzzy logic


A fuzzy set A fuzzy set (A generalized concept of the (A generalized concept of the conventional crisp set)conventional crisp set) is specified with a is specified with a membership functionmembership function AA(x) (x) which which represents degree of membership in the represents degree of membership in the set.set.

A ={ (x,A ={ (x,AA(x)) | x(x)) | xU , 0U , 0 AA(x) (x) 1} 1}


Basic Set-Theoretic Basic Set-Theoretic OperationsOperationsBasic Set-Theoretic Basic Set-Theoretic OperationsOperations

Equality:Equality: Subset:Subset: Complement:Complement: Union:Union:

Intersection:Intersection:

A B A B

C A B x x x x xc A B A B ( ) max( ( ), ( )) ( ) ( )

C A B x x x x xc A B A B ( ) min( ( ), ( )) ( ) ( )

A X A x xA A ( ) ( )1

Slides for fuzzy setsSlides for fuzzy sets, J.-s. Roger , J.-s. Roger JangJang

BABA


De Morgan’s LawsDe Morgan’s LawsSet operations on fuzzy sets are supposed to be defined such that the more previous known laws and equalities remain true in fuzzy sets as well.Having the previous definitions for fuzzy set operations, we can verify De Morgan laws in fuzzy logic as well.

BABA BABA Exercise: Prove the above equalities


Fuzzy Set operations in detailsFuzzy Set operations in details

Fuzzy complement is actually a function say c that maps the membership function to the membership function of the complement set .

- Fuzzy - Fuzzy ComplementComplement

)(xA)(xA

)()]([)( xxccx AAA Definition: Any function c :[0,1][0,1] that satisfies the following Axioms c1 and c2 is called a fuzzy complement


RequirementsRequirements Axiom c1. Axiom c1.

(boundary conditions)(boundary conditions) Axiom c2. Axiom c2. (non-increasing condition)(non-increasing condition)

0)1(,1)0( cc

)()(]1,0[, bcacbaifba

Axiom c1 requires that if an element belongs to a fuzzy set to degree zero (one), then it should belong to the complement of this fuzzy set to degree one (zero).

Axiom c2 means that an increase in membership value of a fuzzy set must result in a decrease or no change in membership value of the complement set


Clearly, in classical crisp logic (where domain of definition of the complement function is {0,1}) there is only one complement function which satisfies the above axioms whereas in fuzzy logic, there are many functions with a domain [0,1] which satisfy the above conditions.Examples of fuzzy complements1. Basic Fuzzy Complement

aacorxxc AA 1][)(1)]([


2. Sugeno class of fuzzy complements

),1(1

1)(

a

aac

For any value of the parameter , a particular fuzzy complement function is obtained

3. Yager class of fuzzy complements

),0()1()(1

aac

For any value of the parameter , a particular fuzzy complement function is obtained


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

landa=-0.8

landa=-0.5

landa=0

landa=2

landa=8

landa=100

Sugeno Class of Fuzzy Complements for different values of Landa

Graphical Representation of the Sugeno Class Complement


Graphical Representation of the Yager Class Complement

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

w=0.3

w=0.5

w=0.7

w=1

w=2

w=7

Yager Class of Fuzzy Complements for different values of w


Intuitively, the union of two sets, AIntuitively, the union of two sets, AB means B means a fuzzy set (in particular the smallest one) a fuzzy set (in particular the smallest one) containing both A and B.containing both A and B.

The union of two fuzzy sets can be defined The union of two fuzzy sets can be defined with a function named s-norm with a function named s-norm s:s:[0,1]x[0,1][0,1]x[0,1][0,1][0,1] which maps the which maps the membership functions of fuzzy sets A and B membership functions of fuzzy sets A and B into the membership function of the union of into the membership function of the union of A and BA and B

(called A(called AB)B)

- Fuzzy Union- Fuzzy Union s-norm (t-s-norm (t-conorm)conorm)

)()](),([ xxxs BABA The requirements for a function to be an s-norm are as follow:


Axiom s1Axiom s1.. (boundary conditions) (boundary conditions)

Axiom s2Axiom s2.. (commutative condition) (commutative condition)

Axiom s3Axiom s3.. (non-decreasing condition) (non-decreasing condition)

Axiom s4Axiom s4.. (associative condition) (associative condition)

aasass )0,(),0(,1)1,1(

),(),( absbas

bbandaaif),(),( basbas

)),(,()),,(( cbsascbass


Definition:Definition:Any function s:[0,1]x[0,1]Any function s:[0,1]x[0,1][0,1] that satisfies [0,1] that satisfies the above 4 axioms is called an s-normthe above 4 axioms is called an s-norm

Examples of fuzzy s-norms

1. Dombi calss

1])1

1()1

1[(1

1),(

ba

bas),0(

2. Dubois-Prade calss

),1,1max(

)1,,min(),(

ba

baabbabas

)1,0(


3. Yager calss])(,1min[),(

1

babas ),0(

4. Drastic Sum:

otherwise

aifb

bifa

basds1

0

0

),(

5. Einstein Sum:

ab

babases

1),(


6. Algebraic Sum: abbabasas ),(

7. Maximum (Basic fuzzy Union)

),max(),(max babas

Theorem S1: For any s-norm, s(a,b) the following inequality holds: (for any a,b [0,1] ),(),(),max( basbasba dsIt means that the smallest s-norm (or smallest union of two fuzzy sets) is maximum while the largest s-norm is Drastic sum


Theorem S2:Theorem S2: Dombi s-norm and Yager s- Dombi s-norm and Yager s-norm cover the whole spectrum of s-norms norm cover the whole spectrum of s-norms when their parameters changewhen their parameters changeIn its extreme In its extreme cases:cases:

So it is possible to build any s-norm with choosing the right parameter in any of the yager or dombi s-norms

)(a,bmax

),( basim ),(),(

0basbasim ds

)(a,bmax

),( basim ),(),(

0basbasim ds

And Also


Intuitively, the intersection of two sets, AIntuitively, the intersection of two sets, AB B means a fuzzy set (in particular the largest means a fuzzy set (in particular the largest one) containing by both A and B.one) containing by both A and B.

The Intersection of two fuzzy sets can be The Intersection of two fuzzy sets can be defined with a function named t-norm defined with a function named t-norm t:t:[0,1]x[0,1][0,1]x[0,1][0,1][0,1] which maps the which maps the membership functions of fuzzy sets A and B membership functions of fuzzy sets A and B into the membership function of the into the membership function of the intersection of intersection of A and BA and B

- Fuzzy Intersection- Fuzzy Intersection t-t-normnorm

)()](),([ xxxt BABA The requirements for a function to be a t-norm are as follow:


Axiom t1Axiom t1.. (boundary conditions) (boundary conditions)

Axiom t2Axiom t2.. (commutative condition) (commutative condition)

Axiom t3Axiom t3.. (non-decreasing condition) (non-decreasing condition)

Axiom t4Axiom t4.. (associative condition) (associative condition)

aatatt ),1()1,(,0)0,0(

),(),( abtbat

bbandaaif),(),( batbat

)),(,()),,(( cbtatcbatt


Definition:Definition:Any function t:[0,1]x[0,1]Any function t:[0,1]x[0,1][0,1] that satisfies [0,1] that satisfies the above 4 axioms is called a t-normthe above 4 axioms is called a t-norm

Examples of fuzzy t-norms

1. Dombi calss

1])1

1()1

1[(1

1),(

ba

bat),0(

2. Dubois-Prade calss

),,max(),(

ba

abbat )1,0(


3. Yager calss

]))1()1((,1min[1),(1

babat

),0(

4. Drastic Product:

otherwise

aifb

bifa

batdp0

1

1

),(

5. Einstein Product:

)(2),(

abba

abbatep


6. Algebraic Product: abbatap ),(

7. Minimum (Basic fuzzy Intersection)

),min(),(min babas

Theorem T1: For any t-norm, t(a,b) the following inequality holds: (for any a,b [0,1] ) ),min(),(),( babatbatdp It means that the largest t-norm (or largest intersection of two fuzzy sets) is minimum while the smallest t-norm is Drastic product


Theorem T2:Theorem T2: Dombi t-norm and Yager t- Dombi t-norm and Yager t-norm cover the whole spectrum of t-norm cover the whole spectrum of t-norms when their parameters changenorms when their parameters change

In its extreme cases:In its extreme cases:

)(a,bmin

),( batim ),(),(

0batbatim dp

And also:And also:

)(a,bmin

),( batim ),(),(

0basbatim dp

So it is possible to build any t-norm with choosing the right parameter in any of the yager or dombi t-norms.


Graphical representation of theorem S1 Graphical representation of theorem S1 and S2 and S2

)(xA)(xB

)(xBA

S(a,b)=max(a,b)

Yager: S(a,b)=sw (a,b) W=3

Algebraic sum: S(a,b)=sas (a,b)

),(),(),max( basbasba ds


Graphical representation of theorem T1 and Graphical representation of theorem T1 and T2 T2

)(xA)(xB

)(xBAS(a,b)=min(a,b)Yager: t(a,b)=tw (a,b) W=3

Algebraic product: S(a,b)=sap (a,b)

),min(),(),( babatbatdp


Generalized De Morgan’s LawGeneralized De Morgan’s LawGeneralized De Morgan’s LawGeneralized De Morgan’s Law Using the new definitions of s-norm and t-norm Using the new definitions of s-norm and t-norm

instead of the basic fuzzy union and basic fuzzy instead of the basic fuzzy union and basic fuzzy intersection respectively, the generalized De Morgan’s intersection respectively, the generalized De Morgan’s Law can be shown as follow:Law can be shown as follow:

c( t(a,b) ) = s( c(a), c(b) )c( t(a,b) ) = s( c(a), c(b) )

c( s(a,b) ) = t( c(a), c(b) )c( s(a,b) ) = t( c(a), c(b) )where c(.) denotes for any fuzzy complement and s(.) where c(.) denotes for any fuzzy complement and s(.) and t(.) denote for fuzzy s-norm and fuzzy t-norm and t(.) denote for fuzzy s-norm and fuzzy t-norm respectively.respectively.


Associated classAssociated classAn s-norm s(a,b), a t-norm t(a,b) and An s-norm s(a,b), a t-norm t(a,b) and a fuzzy complement c(a) form an a fuzzy complement c(a) form an associated classassociated class if they all together if they all together satisfy the Generalized De Morgan satisfy the Generalized De Morgan lawslaws

c[s(a,b)]=t[c(a),c(b)]c[s(a,b)]=t[c(a),c(b)]It can be shown that there is a t-norm associated It can be shown that there is a t-norm associated with each s-norm in the sense that there is a with each s-norm in the sense that there is a complement such that the De Morgan laws are complement such that the De Morgan laws are satisfied. For example the Yager s-norm and t-satisfied. For example the Yager s-norm and t-norms are associated with each other through basic norms are associated with each other through basic fuzzy complementfuzzy complement


ReferencesReferences 1. L.X. Wang, A course in Fuzzy Systems 1. L.X. Wang, A course in Fuzzy Systems

and controland control 2. 2. Tutorial on Fuzzy LogicTutorial on Fuzzy Logic, Jan Jantzen, , Jan Jantzen,

Technical University of Denmark, Technical University of Denmark, Technical report no 98-E 868, 1999Technical report no 98-E 868, 1999

3. 3. Slides for fuzzy setsSlides for fuzzy sets, J.-s. Roger Jang , J.-s. Roger Jang http://http://www.cs.nthu.edu.tw/~jangwww.cs.nthu.edu.tw/~jang

Documents

Lecture 4 (Fuzzy Set Operations)